This is the actual provincial contest from 2009. For an extra challenge, work on these questions and post solutions below. Please share any tricks or shortcuts you used!! Don't limit yourself to plain text - enhance with TeX, pictures, links, etc.
Blitz 3.
n - number of each coin
0.05n + 0.10 n + 0.25n +1.00 n = $1.40 n
max multiple under $20 is $19.60 (1.4 x 14)
Blitz 4.
Figure out how many ways the friends can sit in 3 seats (ignoring order for now): 3 ways
Next, figure out how many ways to order the 3 friends in their seats: 3! = 6
Total ways: 3 x 6 = 18
Blitz 5.
L - length, 1000 / L - width
new area = (L)(1.1)(1000/L)(0.9) = 990
Blitz 6.
32 - 16 + 8 - 4 + 2 - 1
16 + 4 + 1
21
Blitz 7.
prob = # ways to roll 6 / # rolls possible
ways to roll 6 --> 5&1, 1&5, or 3&3 --> 3
possibilities --> 6x6 --> 36
prob = 1 / 12
Blitz 8.
Blitz 9.
Less than sqrt(2025) = sqrt(25 x 81) = 45
Try 44 ^2 = 1600 + 320 + 16 = 1936
sqrt(2009) is somewhere between 44 and 45, closer to 45.
Blitz 10.
100 = 2 x 2 x 5 x 5, so factorization of n! must have two 2's (4!) and two 5's (10!)
n = 10
Blitz 12
13x6=78
78+3=81
$81
Blitz 15
If 20 is both the average, and the middle number of the set, the numbers could be 1, 2, 20, 38, 39
Bull's Eye Section
Bull's-eye 1
There are an equal number of bicycles as there are tricycles, so we can count 1 bicycle and one tricycle as a set (5 wheels). There are 330 wheels in total, so the number of bicycles/tricycles is 330/5, or 66
Bull's Eye 5
Just list all the factors, then add: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
the sum is 168
Bull's Eye 6
correct ways to line up = 4 (Middle must be AB or BA, outsides must be CD or DC 2x2=4)
total ways to line up = 4!
Prob = 4 / 4! = 1 / 6
Bull's Eye7
we can call the 4 numbers (from smallest to biggest) S, a, c, and B.
if we want S+B to be as small as possible, S should be 1, and a+c+B=299
a, c, and B should be as close to each other as possible, so therefore B= 101, a= 100, and c= 98
101+ 1=102
Bull's Eye 8 Handshake counting strategy : add up each person's shakes, then divide total by 2 since you will have counted each handshake twice
The 11 people that shake everyone's hand shake 19 hands. 11 x 19 = 209
The 9 people that shake 13 friends' hands shake 13 hands. 9 x 13 = 117
Total shakes = (209 + 117) / 2 = 163
Bull's Eye 9
A Regular hexagon has 6 lines of symmetry (just draw one, and count)
Hexagon symmetry
Bull's eye 4
We can divide the candles into units, each x cm tall
The short thick candle burns one of these units every 30 minutes (there are 4 units in total in the short, thick candle)
we know that after thirty minutes, there are 3 units left in the short, thick candle, and since both candles are equal in height, there are 3 units left in the originally tall candle
after another thirty minutes, the originally short, thick candle has two units left, and since the originally tall candle is half the height of the originally short candle, it has one unit left.
using this, we can calculate that the originally tall candle burns two units every 30 minutes, and one unit every 15 minutes
The tall candle was already burning for 60 minutes, and since there is one unit left, the candle still has 15 more minutes until it burns out
The total expected burn time of the originally tall candle is 75 minutes
Co-op Section
Co-op 1
Guess and check: 1 and 2 dont work, 2 and 3 don't work, etc. until 14 (1, 2, 7, 14) and 15 (1, 3, 5, 15)
TL- Just a tip, you can eliminate 1, prime numbers, and numbers that are squared from others (eg. 4 and 9)
Co-op 2
6!= 1*2*3*4*5*6
=5*3*2 factors
=30
For 7!, the factors would be:
which means: 5*3*2*2 factors
so the factors of 6!/7!
=30/60
=1/2
Great! With that analysis in mind, you could solve the question even faster by realizing that
7 x 6 x 5 x 4 x 3 x 2 x 1 must have double the # divisors as 6 x 5 x 4 x 3 x 2 x 1
since it would have all the same divisors AND all the same divisors times 7.
Co-op 8.
Work backwards.
5 could come from 50, 14, 41, 23, 32
50, 41, 23, 32 can't go back another step (only numbers 18 or less could go back two steps)
14 could come from 95, 59, 86, 68, 77
So there are 10 two-digit numbers .
Co-op 9.
There will be 3 occurrences between 1PM and 4PM when the minute and hour hands are separated by 180 degrees (the first sometime between 1:00 and 2:00, the second between 2:00 and 3:00, and the third between 3:00 and 4:00).
Let's consider the first occurrence which occurs between 1:00 and 2:00.
Minute Position - Hour Position = 180
360 t - (30 + 30t) = 180
..solve..
t = 7 / 11 (1:00 to 2:00)
So time = 1 + 7/11
Similar analysis for 2:00 to 3:00
360 t - (60 + 30t) = 180
..solve..
t = 8 / 11 (2:00 to 3:00)
So time = 2 + 8/11
Similar analysis for 2:00 to 3:00
360 t - (90 + 30t) = 180
..solve..
t = 9 / 11 (3:00 to 4:00)
So time = 3 + 9/11
Add up the three times:
1 + 7/11+ 2 + 8/11+ 3 + 9/11
= 6 + 24 / 11
= (66 + 24) / 11
= 90 / 11
Co-op 10.
First, let's add subscripts to the the students to indicate their original group.
We need to rearrange them so each new group doesn't have the same subscript twice. In other words, each new group will have 3 of the 4 subscripts. The new groups will look like this:
For each X_1, we must fill in A, B, or C. The number of ways to do this is 3 x 2 x 1 =6 ways
For each X_2, we must fill in D, E, or F. There are 6 ways to do this.
For each X_3, we must fill in G, H, or I. There are 6 ways to do this.
For each X_4, we must fill in J, K, or L. There are 6 ways to do this.
So the total number of ways is
Face-Off Section
Face off 1
Fraction over fraction, so multiply top by 100
25 + 4
29
Face off 2
The rectangle has a perimeter 3, and there are 12 different sides of squares (congruent) making up the perimeter, therefore, each square's side is 3/12
The area of the rectangle would be:
15/12 x 3/12
=15/4 x 1/12
=15/48 cm squared
Here's another solution:
So x = 1/4 cm
Area = 5/16 cm squared
Face off 3
(n+2)(n-2) = 50 (n-2)
Compare both sides..
(n+2) = 50
n = 48
Face off 4.
14 into 168 = 12 (easy with short division if you don't have it memorized)
16.8 - decimal left 1
0.014 - decimal right 3
So 1200
Face off 5.
12 min x 2.25 = 27 min
1000 m x 2.25 = 2250 m
Face off 6
7*7=49 combinations
Face off 31 8! .
5!*3!
=6*7*8
2*3
=7*8
=56
Face off 8:
to find the total number of punt yards, use the equation 3*35=105
105-41 (the longest punt) =64
64/2=32
average of the other two punts =32 yards
Face off 11
Guess and check:
182/4= about 45
44+45+46+47=182
largest integer=47
Face off 18
3 x $2.99= 9 - .03 = $8.97
2 x $3.99= 8 - .02 = $7.98
$8.79 - $7.98 = $.81
81 cents
Face off 13
the surface area of the cube is equal to 54 units squared
when you take away those gray squares, you add 9 extra units squared to the surface area
59 + 9 = 68
Face off 15
300 kids have MP3's, 250 have cells, and 200 have both. To find how many do not have either, the equation is 500 - (300 + 50) because 200 of the kids who have cell phones have MP3's, and were counted already.
Blitz Section
Blitz 1.Fraction over a fraction - Flip and multiply denominator
=(1/2 + 3/4 + 5/6)(12/5)
=(6 + 9 + 10) / 5
=5
Blitz 2.
=(2/7)(60 seconds)
=120 / 7 ...short division.. 17.1...
=17
Blitz 3.
n - number of each coin
0.05n + 0.10 n + 0.25n +1.00 n = $1.40 n
max multiple under $20 is $19.60 (1.4 x 14)
Blitz 4.
- Figure out how many ways the friends can sit in 3 seats (ignoring order for now): 3 ways
- Next, figure out how many ways to order the 3 friends in their seats: 3! = 6
Total ways: 3 x 6 = 18Blitz 5.
L - length, 1000 / L - width
new area = (L)(1.1)(1000/L)(0.9) = 990
Blitz 6.
32 - 16 + 8 - 4 + 2 - 1
16 + 4 + 1
21
Blitz 7.
prob = # ways to roll 6 / # rolls possible
- ways to roll 6 --> 5&1, 1&5, or 3&3 --> 3
- possibilities --> 6x6 --> 36
prob = 1 / 12Blitz 8.
Blitz 9.
Less than sqrt(2025) = sqrt(25 x 81) = 45
Try 44 ^2 = 1600 + 320 + 16 = 1936
sqrt(2009) is somewhere between 44 and 45, closer to 45.
Blitz 10.
100 = 2 x 2 x 5 x 5, so factorization of n! must have two 2's (4!) and two 5's (10!)
n = 10
Blitz 12
13x6=78
78+3=81
$81
Blitz 15
If 20 is both the average, and the middle number of the set, the numbers could be 1, 2, 20, 38, 39
Bull's Eye Section
Bull's-eye 1
There are an equal number of bicycles as there are tricycles, so we can count 1 bicycle and one tricycle as a set (5 wheels). There are 330 wheels in total, so the number of bicycles/tricycles is 330/5, or 66
Bull's Eye 5
Just list all the factors, then add: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
the sum is 168
Bull's Eye 6
Prob = 4 / 4! = 1 / 6
Bull's Eye7
we can call the 4 numbers (from smallest to biggest) S, a, c, and B.
if we want S+B to be as small as possible, S should be 1, and a+c+B=299
a, c, and B should be as close to each other as possible, so therefore B= 101, a= 100, and c= 98
101+ 1=102
Bull's Eye 8
Handshake counting strategy : add up each person's shakes, then divide total by 2 since you will have counted each handshake twice
The 11 people that shake everyone's hand shake 19 hands. 11 x 19 = 209
The 9 people that shake 13 friends' hands shake 13 hands. 9 x 13 = 117
Total shakes = (209 + 117) / 2 = 163
Bull's Eye 9
A Regular hexagon has 6 lines of symmetry (just draw one, and count)
Bull's eye 4
We can divide the candles into units, each x cm tall
The short thick candle burns one of these units every 30 minutes (there are 4 units in total in the short, thick candle)
we know that after thirty minutes, there are 3 units left in the short, thick candle, and since both candles are equal in height, there are 3 units left in the originally tall candle
after another thirty minutes, the originally short, thick candle has two units left, and since the originally tall candle is half the height of the originally short candle, it has one unit left.
using this, we can calculate that the originally tall candle burns two units every 30 minutes, and one unit every 15 minutes
The tall candle was already burning for 60 minutes, and since there is one unit left, the candle still has 15 more minutes until it burns out
The total expected burn time of the originally tall candle is 75 minutes
Co-op Section
Co-op 1
Guess and check: 1 and 2 dont work, 2 and 3 don't work, etc. until 14 (1, 2, 7, 14) and 15 (1, 3, 5, 15)
TL- Just a tip, you can eliminate 1, prime numbers, and numbers that are squared from others (eg. 4 and 9)
Co-op 2
6!= 1*2*3*4*5*6
=5*3*2 factors
=30
For 7!, the factors would be:
which means: 5*3*2*2 factors
so the factors of 6!/7!
=30/60
=1/2
Great! With that analysis in mind, you could solve the question even faster by realizing that
7 x 6 x 5 x 4 x 3 x 2 x 1 must have double the # divisors as 6 x 5 x 4 x 3 x 2 x 1
since it would have all the same divisors AND all the same divisors times 7.
Co-op 8.
Work backwards.
5 could come from 50, 14, 41, 23, 32
50, 41, 23, 32 can't go back another step (only numbers 18 or less could go back two steps)
14 could come from 95, 59, 86, 68, 77
So there are 10 two-digit numbers .
Co-op 9.
There will be 3 occurrences between 1PM and 4PM when the minute and hour hands are separated by 180 degrees (the first sometime between 1:00 and 2:00, the second between 2:00 and 3:00, and the third between 3:00 and 4:00).
Let's consider the first occurrence which occurs between 1:00 and 2:00.
Minute Position - Hour Position = 180
360 t - (30 + 30t) = 180
..solve..
t = 7 / 11 (1:00 to 2:00)
So time = 1 + 7/11
Similar analysis for 2:00 to 3:00
360 t - (60 + 30t) = 180
..solve..
t = 8 / 11 (2:00 to 3:00)
So time = 2 + 8/11
Similar analysis for 2:00 to 3:00
360 t - (90 + 30t) = 180
..solve..
t = 9 / 11 (3:00 to 4:00)
So time = 3 + 9/11
Add up the three times:
1 + 7/11+ 2 + 8/11+ 3 + 9/11
= 6 + 24 / 11
= (66 + 24) / 11
= 90 / 11
Co-op 10.
First, let's add subscripts to the the students to indicate their original group.
We need to rearrange them so each new group doesn't have the same subscript twice. In other words, each new group will have 3 of the 4 subscripts. The new groups will look like this:
For each X_1, we must fill in A, B, or C. The number of ways to do this is 3 x 2 x 1 =6 ways
For each X_2, we must fill in D, E, or F. There are 6 ways to do this.
For each X_3, we must fill in G, H, or I. There are 6 ways to do this.
For each X_4, we must fill in J, K, or L. There are 6 ways to do this.
So the total number of ways is
Face-Off Section
Face off 1Fraction over fraction, so multiply top by 100
25 + 4
29
Face off 2
The rectangle has a perimeter 3, and there are 12 different sides of squares (congruent) making up the perimeter, therefore, each square's side is 3/12
The area of the rectangle would be:
15/12 x 3/12
=15/4 x 1/12
=15/48 cm squared
Here's another solution:
So x = 1/4 cm
Area = 5/16 cm squared
Face off 3
(n+2)(n-2) = 50 (n-2)
Compare both sides..
(n+2) = 50
n = 48
Face off 4.
14 into 168 = 12 (easy with short division if you don't have it memorized)
16.8 - decimal left 1
0.014 - decimal right 3
So 1200
Face off 5.
12 min x 2.25 = 27 min
1000 m x 2.25 = 2250 m
Face off 6
7*7=49 combinations
Face off 31
8! .
5!*3!
=6*7*8
2*3
=7*8
=56
Face off 8:
to find the total number of punt yards, use the equation 3*35=105
105-41 (the longest punt) =64
64/2=32
average of the other two punts =32 yards
Face off 11
Guess and check:
182/4= about 45
44+45+46+47=182
largest integer=47
Face off 18
3 x $2.99= 9 - .03 = $8.97
2 x $3.99= 8 - .02 = $7.98
$8.79 - $7.98 = $.81
81 cents
Face off 13
the surface area of the cube is equal to 54 units squared
when you take away those gray squares, you add 9 extra units squared to the surface area
59 + 9 = 68
Face off 15
300 kids have MP3's, 250 have cells, and 200 have both. To find how many do not have either, the equation is 500 - (300 + 50) because 200 of the kids who have cell phones have MP3's, and were counted already.
500 - 350= 150 kids